A system of inequalities can be used to determine the depth of a toy, in meters, in a pool depending on the time, in seconds, since it was dropped. Which constraint could be part of the scenario?
A. The pool is 1 meter deep.
B. The pool is 2 meters deep.
C. The toy falls at a rate of at least a [tex]$\frac{1}{2}$[/tex] meter per second.
D. The toy sinks at a rate of no more than a [tex]$\frac{1}{2}$[/tex] meter per second.
Answer (1)
The toy sinks at a rate of no more than a 2 1 meter per second.
Explanation
Analyze the options Let's analyze the given options to determine which constraint could be part of the scenario.
Option 1: The pool is 1 meter deep. This implies that the depth of the toy, d , must be less than or equal to 1 meter. So, d l e 1 .
Option 2: The pool is 2 meters deep. This implies that the depth of the toy, d , must be less than or equal to 2 meters. So, d l e 2 .
Option 3: The toy falls at a rate of at least a 2 1 meter per second. This means the rate of change of depth with respect to time, t d , is greater than or equal to 2 1 . So, t d g e 2 1 , which means d g e 2 1 t .
Option 4: The toy sinks at a rate of no more than a 2 1 meter per second. This means the rate of change of depth with respect to time, t d , is less than or equal to 2 1 . So, t d l e 2 1 , which means d l e 2 1 t .
All the options can be part of the scenario. The question asks which constraint could be part of the scenario. Since all options can be expressed as inequalities, any of them is a valid answer. However, we need to choose only one option. Let's analyze the options again.
Options 1 and 2 are similar, they both limit the depth of the pool. Options 3 and 4 describe the rate at which the toy sinks. Since the question mentions a system of inequalities to determine the depth of the toy depending on the time , the options that relate depth and time are more relevant. Between options 3 and 4, option 4 seems more realistic, as the toy's sinking rate is unlikely to increase indefinitely.
Choose the most realistic constraint The problem states that a system of inequalities can be used to determine the depth of a toy in a pool depending on the time since it was dropped. We need to identify which of the given constraints could be part of this scenario. The key here is to recognize that the depth of the toy is related to the time it has been falling. Therefore, options that describe the relationship between depth and time are more likely to be part of the system of inequalities.
Option 3 states that the toy falls at a rate of at least 2 1 meter per second. This means that the depth d is greater than or equal to 2 1 t , where t is the time in seconds. This can be written as d ≥ 2 1 t .
Option 4 states that the toy sinks at a rate of no more than 2 1 meter per second. This means that the depth d is less than or equal to 2 1 t , where t is the time in seconds. This can be written as d ≤ 2 1 t .
Since the toy is sinking, it is more realistic to assume that its rate of sinking is limited. Therefore, option 4 is a more suitable constraint for the scenario.
Final Answer The most realistic constraint is that the toy sinks at a rate of no more than a 2 1 meter per second.
Examples
Imagine you're designing a game where a character falls into water. You can use a system of inequalities to model the character's depth over time. One constraint could be the maximum sinking speed, ensuring the character doesn't fall too fast. This helps create realistic and balanced gameplay.
